Problem: Solve for $x$ : $ 8|x - 4| + 3 = 2|x - 4| + 9 $
Solution: Subtract $ {2|x - 4|} $ from both sides: $ \begin{eqnarray} 8|x - 4| + 3 &=& 2|x - 4| + 9 \\ \\ { - 2|x - 4|} && { - 2|x - 4|} \\ \\ 6|x - 4| + 3 &=& 9 \end{eqnarray} $ Subtract ${3}$ from both sides: $ \begin{eqnarray} 6|x - 4| + 3 &=& 9 \\ \\ { - 3} &=& { - 3} \\ \\ 6|x - 4| &=& 6 \end{eqnarray} $ Divide both sides by ${6}$ $ \dfrac{6|x - 4|} {{6}} = \dfrac{6} {{6}} $ Simplify: $ |x - 4| = 1$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 4 = -1 $ or $ x - 4 = 1 $ Solve for the solution where $x - 4$ is negative: $ x - 4 = -1 $ Add ${4}$ to both sides: $ \begin{eqnarray} x - 4 &=& -1 \\ \\ {+ 4} && {+ 4} \\ \\ x &=& -1 + 4 \end{eqnarray} $ $ x = 3 $ Then calculate the solution where $x - 4$ is positive: $ x - 4 = 1 $ Add ${4}$ to both sides: $ \begin{eqnarray} x - 4 &=& 1 \\ \\ {+ 4} && {+ 4} \\ \\ x &=& 1 + 4 \end{eqnarray} $ $ x = 5 $ Thus, the correct answer is $x = 3 $ or $x = 5 $.